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<H3><A NAME="SECTION03235300000000000000"></A><A NAME="sectionGSVDdriver"></A>
<BR>
Generalized Singular Value Decomposition (GSVD)
</H3>

<P>
<A NAME="1812"></A><A NAME="1813"></A>
<A NAME="1814"></A>
<A NAME="1815"></A>
<A NAME="1816"></A>
The <B>generalized (or quotient) singular value decomposition</B>
of an <I>m</I>-by-<I>n</I> matrix <I>A</I> and a
<I>p</I>-by-<I>n</I> matrix <I>B</I> is given by the pair of factorizations
<BR><P></P>
<DIV ALIGN="CENTER">

<!-- MATH
 \begin{displaymath}
A = U \Sigma_1 [0,\; R ] Q^T
\;\;\; {\rm and} \;\;\;
B = V \Sigma_2 [0, \; R ] Q^T \;.
\end{displaymath}
 -->


<IMG
 WIDTH="345" HEIGHT="31" BORDER="0"
 SRC="img65.gif"
 ALT="\begin{displaymath}
A = U \Sigma_1 [0,\; R ] Q^T
\;\;\; {\rm and} \;\;\;
B = V \Sigma_2 [0, \; R ] Q^T \;.
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
The matrices in these factorizations have the following properties:

<UL><LI><I>U</I> is <I>m</I>-by-<I>m</I>, V is <I>p</I>-by-<I>p</I>, <I>Q</I> is <I>n</I>-by-<I>n</I>, and
all three matrices are orthogonal. If <I>A</I> and
<I>B</I> are complex, these matrices are unitary instead of
orthogonal, and <I>Q</I><SUP><I>T</I></SUP> should be
replaced by <I>Q</I><SUP><I>H</I></SUP> in the pair of factorizations.

<LI><I>R</I> is <I>r</I>-by-<I>r</I>, upper triangular and nonsingular.
[0,<I>R</I>] is <I>r</I>-by-<I>n</I> (in other words, the 0 is an <I>r</I>-by-<I>n</I>-<I>r</I>
zero matrix).
The integer <I>r</I> is the rank of

<!-- MATH
 $\left( \begin{array}{c} A \\B \end{array} \right)$
 -->
<IMG
 WIDTH="60" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="img19.gif"
 ALT="$ \left( \begin{array}{c}
A \\
B
\end{array} \right) $">,
and satisfies <IMG
 WIDTH="46" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="img66.gif"
 ALT="$r \leq n$">.

<LI><IMG
 WIDTH="25" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img67.gif"
 ALT="$\Sigma_1$">
is <I>m</I>-by-<I>r</I>,
<IMG
 WIDTH="25" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img68.gif"
 ALT="$\Sigma_2$">
is <I>p</I>-by-<I>r</I>, both are real, nonnegative  and diagonal, and

<!-- MATH
 $\Sigma_1^T \Sigma_1 + \Sigma_2^T \Sigma_2 = I$
 -->
<IMG
 WIDTH="145" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="img69.gif"
 ALT="$\Sigma_1^T \Sigma_1 + \Sigma_2^T \Sigma_2 = I$">.
Write

<!-- MATH
 $\Sigma_1^T \Sigma_1 = {\rm diag} ( \alpha_1^2 , \ldots , \alpha_r^2 )$
 -->
<IMG
 WIDTH="193" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="img70.gif"
 ALT="$\Sigma_1^T \Sigma_1 = {\rm diag} ( \alpha_1^2 , \ldots , \alpha_r^2 )$">
and

<!-- MATH
 $\Sigma_2^T \Sigma_2 = {\rm diag} ( \beta_1^2 , \ldots , \beta_r^2 )$
 -->
<IMG
 WIDTH="192" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="img71.gif"
 ALT="$\Sigma_2^T \Sigma_2 = {\rm diag} ( \beta_1^2 , \ldots , \beta_r^2 )$">,
where <IMG
 WIDTH="21" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="img72.gif"
 ALT="$\alpha_i$">
and <IMG
 WIDTH="20" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img73.gif"
 ALT="$\beta_i$">
lie in the interval from 0 to 1.
The ratios

<!-- MATH
 $\alpha_1 / \beta_1 , \ldots ,  \alpha_r / \beta_r$
 -->
<IMG
 WIDTH="132" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="img74.gif"
 ALT="$\alpha_1 / \beta_1 , \ldots , \alpha_r / \beta_r$">
are called the <B>generalized singular values</B> of the pair <I>A</I>, <I>B</I>.
If <IMG
 WIDTH="52" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img75.gif"
 ALT="$\beta_i = 0$">,
then the generalized singular value
<A NAME="1824"></A>
<A NAME="1825"></A>

<!-- MATH
 $\alpha_i / \beta_i$
 -->
<IMG
 WIDTH="45" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="img76.gif"
 ALT="$\alpha_i / \beta_i$">
is <B>infinite</B>.

</UL>

<P>
<IMG
 WIDTH="25" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img67.gif"
 ALT="$\Sigma_1$">
and <IMG
 WIDTH="25" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img68.gif"
 ALT="$\Sigma_2$">
have the following detailed
structures, depending on whether <IMG
 WIDTH="81" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="img77.gif"
 ALT="$m-r \geq 0$">
or
<I>m</I>-<I>r</I> &lt; 0. In the first case, <IMG
 WIDTH="81" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="img77.gif"
 ALT="$m-r \geq 0$">,
then
<BR><P></P>
<DIV ALIGN="CENTER">

<!-- MATH
 \begin{displaymath}
\Sigma_1 = \bordermatrix{ & k & l \cr
                 \hfill k & I & 0 \cr
                 \hfill l & 0 & C \cr
                    m-k-l & 0 & 0 }
                 \; \; \; {\rm and} \; \; \;
\Sigma_2 = \bordermatrix{ & k & l \cr
                 \hfill l & 0 & S \cr
                      p-l & 0 & 0 } \; .
\end{displaymath}
 -->


<IMG
 WIDTH="421" HEIGHT="98" BORDER="0"
 SRC="img78.gif"
 ALT="\begin{displaymath}
\Sigma_1 = \bordermatrix{ &amp; k &amp; l \cr
\hfill k &amp; I &amp; 0 \cr
...
...rmatrix{ &amp; k &amp; l \cr
\hfill l &amp; 0 &amp; S \cr
p-l &amp; 0 &amp; 0 } \; .
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
Here <I>l</I> is the rank of <I>B</I>, <I>k</I>=<I>r</I>-<I>l</I>, <I>C</I> and <I>S</I> are diagonal
matrices satisfying 
<!-- MATH
 $C^2  + S^2 = I$
 -->
<I>C</I><SUP>2</SUP>  + <I>S</I><SUP>2</SUP> = <I>I</I>, and <I>S</I> is nonsingular.
We may also identify

<!-- MATH
 $\alpha_1 = \cdots = \alpha_k = 1$
 -->
<IMG
 WIDTH="141" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="img79.gif"
 ALT="$\alpha_1 = \cdots = \alpha_k = 1$">,

<!-- MATH
 $\alpha_{k+i} = c_{ii}$
 -->
<IMG
 WIDTH="80" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="img80.gif"
 ALT="$\alpha_{k+i} = c_{ii}$">
for 
<!-- MATH
 $i=1, \ldots , l$
 -->
<IMG
 WIDTH="87" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img81.gif"
 ALT="$i=1, \ldots , l$">,

<!-- MATH
 $\beta_1 = \cdots = \beta_k = 0$
 -->
<IMG
 WIDTH="139" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img82.gif"
 ALT="$\beta_1 = \cdots = \beta_k = 0$">,
and

<!-- MATH
 $\beta_{k+i} = s_{ii}$
 -->
<IMG
 WIDTH="79" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img83.gif"
 ALT="$\beta_{k+i} = s_{ii}$">
for 
<!-- MATH
 $i=1, \ldots , l$
 -->
<IMG
 WIDTH="87" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img81.gif"
 ALT="$i=1, \ldots , l$">.
Thus, the first <I>k</I> generalized singular values

<!-- MATH
 $\alpha_1 / \beta_1 , \ldots , \alpha_k / \beta_k$
 -->
<IMG
 WIDTH="134" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="img84.gif"
 ALT="$\alpha_1 / \beta_1 , \ldots , \alpha_k / \beta_k$">
are infinite, and the remaining <I>l</I> generalized singular values
are finite.

<P>
In the second case, when <I>m</I>-<I>r</I> &lt; 0,
<BR><P></P>
<DIV ALIGN="CENTER">

<!-- MATH
 \begin{displaymath}
\Sigma_1 = \bordermatrix{ & k & m-k & k+l-m \cr
                 \hfill k & I &  0  &   0   \cr
                      m-k & 0 &  C  &   0   }
\end{displaymath}
 -->


<IMG
 WIDTH="288" HEIGHT="79" BORDER="0"
 SRC="img85.gif"
 ALT="\begin{displaymath}
\Sigma_1 = \bordermatrix{ &amp; k &amp; m-k &amp; k+l-m \cr
\hfill k &amp; I &amp; 0 &amp; 0 \cr
m-k &amp; 0 &amp; C &amp; 0 }
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
and
<BR><P></P>
<DIV ALIGN="CENTER">

<!-- MATH
 \begin{displaymath}
\Sigma_2 = \bordermatrix{ & k & m-k & k+l-m \cr
               \hfill m-k & 0 &  S  &   0   \cr
                    k+l-m & 0 &  0  &   I   \cr
               \hfill p-l & 0 &  0  &   0   } \; .
\end{displaymath}
 -->


<IMG
 WIDTH="329" HEIGHT="98" BORDER="0"
 SRC="img86.gif"
 ALT="\begin{displaymath}
\Sigma_2 = \bordermatrix{ &amp; k &amp; m-k &amp; k+l-m \cr
\hfill m-k ...
... &amp; 0 \cr
k+l-m &amp; 0 &amp; 0 &amp; I \cr
\hfill p-l &amp; 0 &amp; 0 &amp; 0 } \; .
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
Again, <I>l</I> is the rank of <I>B</I>, <I>k</I>=<I>r</I>-<I>l</I>, <I>C</I> and <I>S</I> are diagonal
matrices satisfying <I>C</I><SUP>2</SUP> + <I>S</I><SUP>2</SUP> = <I>I</I>, <I>S</I> is nonsingular,
and we may identify

<!-- MATH
 $\alpha_1 = \cdots = \alpha_k = 1$
 -->
<IMG
 WIDTH="141" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="img79.gif"
 ALT="$\alpha_1 = \cdots = \alpha_k = 1$">,

<!-- MATH
 $\alpha_{k+i} = c_{ii}$
 -->
<IMG
 WIDTH="80" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="img80.gif"
 ALT="$\alpha_{k+i} = c_{ii}$">
for 
<!-- MATH
 $i=1, \ldots , m-k$
 -->
<IMG
 WIDTH="127" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img87.gif"
 ALT="$i=1, \ldots , m-k$">,

<!-- MATH
 $\alpha_{m+1} = \cdots = \alpha_r = 0$
 -->
<IMG
 WIDTH="163" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="img88.gif"
 ALT="$\alpha_{m+1} = \cdots = \alpha_r = 0$">,

<!-- MATH
 $\beta_1 = \cdots = \beta_k = 0$
 -->
<IMG
 WIDTH="139" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img82.gif"
 ALT="$\beta_1 = \cdots = \beta_k = 0$">,

<!-- MATH
 $\beta_{k+i} = s_{ii}$
 -->
<IMG
 WIDTH="79" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img83.gif"
 ALT="$\beta_{k+i} = s_{ii}$">
for 
<!-- MATH
 $i=1, \ldots , m-k$
 -->
<IMG
 WIDTH="127" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img87.gif"
 ALT="$i=1, \ldots , m-k$">,
and

<!-- MATH
 $\beta_{m+1} = \cdots = \beta_r = 1$
 -->
<IMG
 WIDTH="160" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img89.gif"
 ALT="$\beta_{m+1} = \cdots = \beta_r = 1$">.
Thus, the first <I>k</I> generalized singular values

<!-- MATH
 $\alpha_1 / \beta_1 , \ldots , \alpha_k / \beta_k$
 -->
<IMG
 WIDTH="134" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="img84.gif"
 ALT="$\alpha_1 / \beta_1 , \ldots , \alpha_k / \beta_k$">
are infinite, and the remaining <I>l</I> generalized singular values
are finite.

<P>
Here are some important special cases of the generalized singular value
decomposition.
<A NAME="1843"></A>
<A NAME="1844"></A>
First, if <I>B</I> is square and nonsingular, then <I>r</I>=<I>n</I> and the
generalized singular value decomposition of <I>A</I> and <I>B</I> is equivalent
to the singular value decomposition of <I>AB</I><SUP>-1</SUP>, where the singular
values of <I>AB</I><SUP>-1</SUP> are equal to the generalized singular values of the
pair <I>A</I>, <I>B</I>:
<BR><P></P>
<DIV ALIGN="CENTER">

<!-- MATH
 \begin{displaymath}
AB^{-1} = (U \Sigma_1 R Q^T)(V \Sigma_2 R Q^T)^{-1} =
U ( \Sigma_1 \Sigma_2^{-1} ) V^T \; \; .
\end{displaymath}
 -->


<IMG
 WIDTH="397" HEIGHT="31" BORDER="0"
 SRC="img90.gif"
 ALT="\begin{displaymath}
AB^{-1} = (U \Sigma_1 R Q^T)(V \Sigma_2 R Q^T)^{-1} =
U ( \Sigma_1 \Sigma_2^{-1} ) V^T \; \; .
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
Second, if
the columns of 
<!-- MATH
 $(A^T \; B^T)^T$
 -->
<IMG
 WIDTH="82" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="img91.gif"
 ALT="$(A^T \; B^T)^T$">
are orthonormal, then <I>r</I>=<I>n</I>, <I>R</I>=<I>I</I> and the
generalized
singular value decomposition of <I>A</I> and <I>B</I> is equivalent to the CS
(Cosine-Sine) decomposition of 
<!-- MATH
 $(A^T \; B^T)^T$
 -->
<IMG
 WIDTH="82" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="img91.gif"
 ALT="$(A^T \; B^T)^T$">
[<A
 HREF="node151.html#GVL2">55</A>]:
<A NAME="1851"></A>
<BR><P></P>
<DIV ALIGN="CENTER">

<!-- MATH
 \begin{displaymath}
\left( \begin{array}{c} A \\B \end{array} \right) = \left( \begin{array}{cc} U & 0 \\0 & V \end{array} \right) 
\left( \begin{array}{c} \Sigma_1 \\\Sigma_2 \end{array} \right) Q^T \; \; .
\end{displaymath}
 -->


<IMG
 WIDTH="265" HEIGHT="54" BORDER="0"
 SRC="img92.gif"
 ALT="\begin{displaymath}
\left( \begin{array}{c} A \\ B \end{array} \right) = \left( ...
...array}{c} \Sigma_1 \\ \Sigma_2 \end{array} \right) Q^T \; \; .
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
Third, the generalized eigenvalues and eigenvectors of 
<!-- MATH
 $A^TA - \lambda B^TB$
 -->
<IMG
 WIDTH="111" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="img93.gif"
 ALT="$A^TA - \lambda B^TB$">
can be expressed in terms of the generalized singular value decomposition:
Let
<BR><P></P>
<DIV ALIGN="CENTER">

<!-- MATH
 \begin{displaymath}
X = Q \left( \begin{array}{cc} I & 0 \\0 & R^{-1} \end{array} \right) \; \; .
\end{displaymath}
 -->


<IMG
 WIDTH="170" HEIGHT="54" BORDER="0"
 SRC="img94.gif"
 ALT="\begin{displaymath}
X = Q \left( \begin{array}{cc} I &amp; 0 \\ 0 &amp; R^{-1} \end{array} \right) \; \; .
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
Then
<BR><P></P>
<DIV ALIGN="CENTER">

<!-- MATH
 \begin{displaymath}
X^T A^T A X = \left( \begin{array}{cc}
                       0 & 0   \\
                       0 & \Sigma^T_1 \Sigma_1
                       \end{array} \right) \;\; {\rm and} \;\;
X^T B^T B X = \left( \begin{array}{cc}
                       0 & 0   \\
                       0 & \Sigma^T_2 \Sigma_2
                       \end{array} \right).
\end{displaymath}
 -->


<IMG
 WIDTH="482" HEIGHT="54" BORDER="0"
 SRC="img95.gif"
 ALT="\begin{displaymath}
X^T A^T A X = \left( \begin{array}{cc}
0 &amp; 0 \\
0 &amp; \Sigm...
...{cc}
0 &amp; 0 \\
0 &amp; \Sigma^T_2 \Sigma_2
\end{array} \right).
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
Therefore, the columns of <I>X</I> are the eigenvectors of

<!-- MATH
 $A^T A - \lambda  B^T B$
 -->
<IMG
 WIDTH="111" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="img93.gif"
 ALT="$A^TA - \lambda B^TB$">,
and the ``nontrivial'' eigenvalues are the
squares of the generalized singular values (see also section&nbsp;<A HREF="node34.html#secGSEP">2.3.5.1</A>).
``Trivial'' eigenvalues
are those corresponding to the leading <I>n</I>-<I>r</I> columns of <I>X</I>,
which span the common null space of <I>A</I><SUP><I>T</I></SUP> <I>A</I> and <I>B</I><SUP><I>T</I></SUP> <I>B</I>.
<A NAME="1865"></A>
<A NAME="1866"></A>
The ``trivial eigenvalues'' are not well defined<A NAME="tex2html383"
 HREF="footnode.html#foot3925"><SUP>2.1</SUP></A>.

<P>
A single driver routine xGGSVD<A NAME="1868"></A><A NAME="1869"></A><A NAME="1870"></A><A NAME="1871"></A> computes the generalized
singular value decomposition<A NAME="1872"></A><A NAME="1873"></A> of <I>A</I> and <I>B</I> (see Table&nbsp;<A HREF="node36.html#tabdrivegeig">2.6</A>).
The method is based on the method described in
[<A
 HREF="node151.html#paige86a">83</A>,<A
 HREF="node151.html#baidemmel92b">10</A>,<A
 HREF="node151.html#baizha93">8</A>].

<P>
<BR>
<DIV ALIGN="CENTER">

<A NAME="tabdrivegeig"></A>
<DIV ALIGN="CENTER">
<A NAME="1877"></A>
<TABLE CELLPADDING=3 BORDER="1">
<CAPTION><STRONG>Table 2.6:</STRONG>
Driver routines for generalized eigenvalue and singular value problems</CAPTION>
<TR><TD ALIGN="CENTER">Type of</TD>
<TD ALIGN="LEFT">Function and storage scheme</TD>
<TD ALIGN="CENTER" COLSPAN=2>Single precision</TD>
<TD ALIGN="CENTER" COLSPAN=2>Double precision</TD>
</TR>
<TR><TD ALIGN="CENTER">problem</TD>
<TD ALIGN="LEFT">&nbsp;</TD>
<TD ALIGN="LEFT">real</TD>
<TD ALIGN="LEFT">complex</TD>
<TD ALIGN="LEFT">real</TD>
<TD ALIGN="LEFT">complex</TD>
</TR>
<TR><TD ALIGN="CENTER">GSEP</TD>
<TD ALIGN="LEFT">simple driver</TD>
<TD ALIGN="LEFT">SSYGV<A NAME="1889"></A></TD>
<TD ALIGN="LEFT">CHEGV<A NAME="1890"></A></TD>
<TD ALIGN="LEFT">DSYGV<A NAME="1891"></A></TD>
<TD ALIGN="LEFT">ZHEGV
<A NAME="1892"></A></TD>
</TR>
<TR><TD ALIGN="CENTER">&nbsp;</TD>
<TD ALIGN="LEFT">divide and conquer driver</TD>
<TD ALIGN="LEFT">SSYGVD<A NAME="1893"></A></TD>
<TD ALIGN="LEFT">CHEGVD<A NAME="1894"></A></TD>
<TD ALIGN="LEFT">DSYGVD<A NAME="1895"></A></TD>
<TD ALIGN="LEFT">ZHEGVD
<A NAME="1896"></A></TD>
</TR>
<TR><TD ALIGN="CENTER">&nbsp;</TD>
<TD ALIGN="LEFT">expert driver</TD>
<TD ALIGN="LEFT">SSYGVX<A NAME="1897"></A></TD>
<TD ALIGN="LEFT">CHEGVX<A NAME="1898"></A></TD>
<TD ALIGN="LEFT">DSYGVX<A NAME="1899"></A></TD>
<TD ALIGN="LEFT">ZHEGVX
<A NAME="1900"></A></TD>
</TR>
<TR><TD ALIGN="CENTER">&nbsp;</TD>
<TD ALIGN="LEFT">simple driver (packed storage)</TD>
<TD ALIGN="LEFT">SSPGV<A NAME="1902"></A></TD>
<TD ALIGN="LEFT">CHPGV<A NAME="1903"></A></TD>
<TD ALIGN="LEFT">DSPGV<A NAME="1904"></A></TD>
<TD ALIGN="LEFT">ZHPGV
<A NAME="1905"></A></TD>
</TR>
<TR><TD ALIGN="CENTER">&nbsp;</TD>
<TD ALIGN="LEFT">divide and conquer driver</TD>
<TD ALIGN="LEFT">SSPGVD<A NAME="1906"></A></TD>
<TD ALIGN="LEFT">CHPGVD<A NAME="1907"></A></TD>
<TD ALIGN="LEFT">DSPGVD<A NAME="1908"></A></TD>
<TD ALIGN="LEFT">ZHPGVD
<A NAME="1909"></A></TD>
</TR>
<TR><TD ALIGN="CENTER">&nbsp;</TD>
<TD ALIGN="LEFT">expert driver</TD>
<TD ALIGN="LEFT">SSPGVX<A NAME="1910"></A></TD>
<TD ALIGN="LEFT">CHPGVX<A NAME="1911"></A></TD>
<TD ALIGN="LEFT">DSPGVX<A NAME="1912"></A></TD>
<TD ALIGN="LEFT">ZHPGVX
<A NAME="1913"></A></TD>
</TR>
<TR><TD ALIGN="CENTER">&nbsp;</TD>
<TD ALIGN="LEFT">simple driver (band matrices)</TD>
<TD ALIGN="LEFT">SSBGV<A NAME="1915"></A></TD>
<TD ALIGN="LEFT">CHBGV<A NAME="1916"></A></TD>
<TD ALIGN="LEFT">DSBGV<A NAME="1917"></A></TD>
<TD ALIGN="LEFT">ZHBGV
<A NAME="1918"></A></TD>
</TR>
<TR><TD ALIGN="CENTER">&nbsp;</TD>
<TD ALIGN="LEFT">divide and conquer driver</TD>
<TD ALIGN="LEFT">SSBGVD<A NAME="1919"></A></TD>
<TD ALIGN="LEFT">CHBGVD<A NAME="1920"></A></TD>
<TD ALIGN="LEFT">DSBGV<A NAME="1921"></A></TD>
<TD ALIGN="LEFT">ZHBGVD
<A NAME="1922"></A></TD>
</TR>
<TR><TD ALIGN="CENTER">&nbsp;</TD>
<TD ALIGN="LEFT">expert driver</TD>
<TD ALIGN="LEFT">SSBGVX<A NAME="1923"></A></TD>
<TD ALIGN="LEFT">CHBGVX<A NAME="1924"></A></TD>
<TD ALIGN="LEFT">DSBGVX<A NAME="1925"></A></TD>
<TD ALIGN="LEFT">ZHBGVX
<A NAME="1926"></A></TD>
</TR>
<TR><TD ALIGN="CENTER">GNEP</TD>
<TD ALIGN="LEFT">simple driver for Schur factorization</TD>
<TD ALIGN="LEFT">SGGES<A NAME="1928"></A></TD>
<TD ALIGN="LEFT">CGGES<A NAME="1929"></A></TD>
<TD ALIGN="LEFT">DGGES<A NAME="1930"></A></TD>
<TD ALIGN="LEFT">ZGGES
<A NAME="1931"></A></TD>
</TR>
<TR><TD ALIGN="CENTER">&nbsp;</TD>
<TD ALIGN="LEFT">expert driver for Schur factorization</TD>
<TD ALIGN="LEFT">SGGESX<A NAME="1932"></A></TD>
<TD ALIGN="LEFT">CGGESX<A NAME="1933"></A></TD>
<TD ALIGN="LEFT">DGGESX<A NAME="1934"></A></TD>
<TD ALIGN="LEFT">ZGGESX
<A NAME="1935"></A></TD>
</TR>
<TR><TD ALIGN="CENTER">&nbsp;</TD>
<TD ALIGN="LEFT">simple driver for eigenvalues/vectors</TD>
<TD ALIGN="LEFT">SGGEV<A NAME="1936"></A></TD>
<TD ALIGN="LEFT">CGGEV<A NAME="1937"></A></TD>
<TD ALIGN="LEFT">DGGEV<A NAME="1938"></A></TD>
<TD ALIGN="LEFT">ZGGEV
<A NAME="1939"></A></TD>
</TR>
<TR><TD ALIGN="CENTER">&nbsp;</TD>
<TD ALIGN="LEFT">expert driver for eigenvalues/vectors</TD>
<TD ALIGN="LEFT">SGGEVX<A NAME="1940"></A></TD>
<TD ALIGN="LEFT">CGGEVX<A NAME="1941"></A></TD>
<TD ALIGN="LEFT">DGGEVX<A NAME="1942"></A></TD>
<TD ALIGN="LEFT">ZGGEVX
<A NAME="1943"></A></TD>
</TR>
<TR><TD ALIGN="CENTER">GSVD</TD>
<TD ALIGN="LEFT">singular values/vectors</TD>
<TD ALIGN="LEFT">SGGSVD<A NAME="1944"></A></TD>
<TD ALIGN="LEFT">CGGSVD<A NAME="1945"></A></TD>
<TD ALIGN="LEFT">DGGSVD<A NAME="1946"></A></TD>
<TD ALIGN="LEFT">ZGGSVD
<A NAME="1947"></A></TD>
</TR>
</TABLE>
</DIV>
</DIV>
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<ADDRESS>
<I>Susan Blackford</I>
<BR><I>1999-10-01</I>
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